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Iterative Rounding and Iterative Relaxation Some of the slides prepared by Lap Chi Lau

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Steiner Network A graph G = (V,E); A connectivity requirement r(u,v) for each pair of vertices u,v. Steiner Network A subgraph H of G which has r(u,v) disjoint paths between each pair u,v. undirected or directed edge or vertex

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Examples of Steiner Network Spanning tree : r(u,v) = 1 for all pairs of vertices. Steiner tree : r(u,v) = 1 for all pair of required vertices. Steiner forest : r(s i,t i ) = 1 for all source sink pairs. k-edge-connected subgraph : r(u,v) = k for all pairs of vertices.

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Survivable Network Design Survivable network design : find a good Steiner network Minimum cost Steiner network: Given a cost c(e) on each edge, find a Steiner network with minimum total cost. e.g. minimum Steiner tree, k-edge-connected subgraph NP-complete…

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Linear Programming Relaxation S uv f(S) := max{ r(u,v) | S separates u and v}. at least r(u,v) edges crossing S number of constraints is exponential. can be handled by Ellipsoid algorithm need a separation oracle

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Approximating Survivable network Design What does not seem to generalize … ? primal-dual algorithm for constrained Steiner forest only logarithmic approx. is known via this approach one shot rounding as in vertex cover - rounding up fractions ¸ ½ set cover – randomized rounding combinatorial algorithms e.g., Steiner tree (reduction to spanning tree in the metric completion graph) New ideas are needed …

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Basic Solutions (1) A basic solution is the unique solution of m linearly independent tight constraints, where m is the number of variables in the LP. A basic solution cannot be represented as a convex sum of feasible solutions

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Basic Solutions (2) An edge of 0, delete it. An edge of 1, pick it. Tight inequalities all come from the connectivity requirements. A basic solution is the unique solution of m linearly independent tight constraints, where m is the number of variables in the LP.

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Petersen Graph: Non-Basic Feasible Solution use every edge to the extent of 1/3. 15 edges with 0 < x(e) < 1 10 tight constraints (at the vertices)

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Basic Feasible Solution black edge – ¼ red edge – ½

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Tight sets

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LP Solutions All 1/3 is a feasible solution. But not a basic feasible solution Thick edges have value 1/2; Thin edges have value 1/4. This is a basic feasible solution. Theorem [Jain 98] Every basic feasible solution has an edge of value at least 1/2 Corollary. There is a 2-approximation algorithm for survivable network design.

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Iterative Rounding Initialization: H =, f = f. While f 0 do: oFind a basic optimal solution, x, of the LP with function f. oAdd an edge with x(e) 1/2 into H. oUpdate f: for every set S, set Output H. By Jains Theorem 0.5 e f(S)=2 f(S)=1 Corollary. This is a 2-approximation algorithm for the minimum cost survivable network problem. The residual problem is feasible.

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Linear Programming Solver Suitable Rounding Procedure Problem Instance Optimal Fractional Solution Integer Solution Linear Programming Solver Problem Instance Optimal Fractional Solution Part Integer Good Part Too much Fractional Residual Problem Typical Rounding: Iterative Rounding:

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Laminar Basis any pair of sets in the basis are either disjoint or contained tree representation

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Weak Supermodularity A weakly supermodular function f satisfies: Example: f(S) := max{ r(u,v) | S separates u and v}. or Weakly supermodular functions are very useful for obtaining a laminar basis Important: when f is updated by setting x(e)=1 for some edges e, it remains weakly supermodular

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Obtaining a Laminar Basis Uncrossing technique : A basic solution is defined by a laminar family of tight connectivity constraints. A B A[BA[B AÅBAÅB Tight constraints:

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Proof of Jains Theorem There are |L| constraints. There are |E| variables. Theorem [Jain]. Every basic solution has an edge with value at least 1/3 Assume every edge has value 0 < x(e) < 1/3. Prove that |E| > |L| by a counting argument. At the beginning we give 2 tokens to each edge, 1 to each endpoint. At the end we redistribute the tokens so that each member in the laminar family has at least 2 tokens, and there are still some tokens left. Then this would imply |E| > |L|

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Induction Basis Assume every edge has value 0 < x(e) < 1/3. 1/4 each leaf set has degree ¸ 4 otherwise 9 e, x(e) · 1/3 each leaf has 4 tokens

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Inductive Step (1) +2 Theorem [Jain]. Every basic solution has an edge with value at least 1/3 Induction Hypothesis: the root has 2 extra tokens (total of 4)

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Inductive Step (2) +2 root has two children – each child can pass up 2 tokens root has 4 tokens root has one child and 2 new edges – child can pass up 2 tokens root has 4 tokens +2

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Inductive Step (3) root has one child root has no new edges but \delta(child) = \delta(root) linear dependence root has one child and only one new edge e (of the two in picture) both root and child are tight f(root) = f(child) + x(e) or f(root) = f(child) - x(e) but 0

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Summary of Jains Algorithm Theorem [Jain]. Every basic solution has an edge with value at least 1/3 Theorem [Jain]. Every basic solution has an edge with value at least ½ Proof is more involved … Open Question: Combinatorial proof ?

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Survivable Network Design Survivable network design : find a good Steiner network Minimum cost Steiner network: Given a cost c(e) on each edge, find a Steiner network with minimum total cost. e.g. minimum spanning tree, minimum Steiner tree Minimum degree Steiner network: Find a Steiner network with minimum maximum degree. e.g. Hamiltonian path, Hamiltonian cycle

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The Problem Statement Goal: to find a good Steiner network w.r.t. to both criteria Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Without degree bounds, this is the minimum cost Steiner network problem. Without cost on edges, this is the minimum degree Steiner network problem.

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Ideal Approximation Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Let OPT(B) be the value of an optimal solution to this problem. Ideally, we would like to return a solution so that: SOL(B) c·OPT(B) However, it cannot be done for any polynomial factor, even for B=2, since this generalizes the minimum cost Hamiltonian path problem.

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Bicriteria Approximation Algorithms This implies a c-approximation for minimum cost Steiner network, and an f(B)-approximation for minimum degree Steiner network. Minimum cost Steiner network with degree constraints: Given a cost c(e) on each edge, find a Steiner network with minimum total cost, so that every vertex has degree at most B. Let OPT(B) be the value of an optimal solution to this problem. A (c,f(B))-approximation algorithm if it returns a solution with SOL(f(B)) c·OPT(B) maximum degree f(B) e.g. f(B)=2B+1

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Minimum Bounded Degree Spanning Trees Theorem [Furer and Raghavachari 92] Given k, there is a polynomial time algorithm which does the following: either the algorithm (i)finds a spanning tree with maximum degree at most k+1. (ii)shows that there is no spanning tree with maximum degree at most k. Theorem [Goemans 06]: Given k, there is a polynomial time algorithm that computes a spanning tree with cost at most OPT(k) and maximum degree at most k+2. uncrossing! (1,B+2)

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Results on Minimum Degree Survivable Networks Minimum costMinimum degreeBicriteria Spanning tree1B+1 [FR](1,B+2) [G] Steiner tree1.55 [RZ]B+1 [FR](O(logn),O(logn)B) Steiner forest2 [AKR]?? k-ec subgraph2 [KV]O(log n)·B [FMZ]? Steiner network2 [Jain]?? Theorem [Lau,Naor,Salavatipour,Singh, STOC 07]: There is a (2,2B+3)-approximation algorithm for the minimum bounded degree Steiner network problem. Corollary : There is a constant factor approximation algorithm for the Minimum Degree Steiner Network problem. (2,2B+3) 2B+3

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Linear Programming Relaxation S uv f(S) := max{ r(u,v) | S separates u and v}. At least r(u,v) edges crossing S Nonuniform degree bounds

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First Try Observation : Half edges are good for degree bounds as well. Initialization: H =, f = f. While f 0 do: oFind a basic optimal solution, x, of the LP with function f. oAdd an edge with x(e) 1/2 into H. oUpdate f: for every set S, set o Update degree bounds : Output H. set B v :=B v -0.5 if e is incident on v. The residual problem is feasible. 0.5 e B v =2 B v =1.5 Problem: A half edge may not exist!

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The Difference But integrality is important in the counting argument. Uncrossing would just work fine. fractional values B v =

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New Idea Idea: Relax the problem by removing the degree constraint for v if v is of low degree. Intuition: Removing a constraint decreases the number of linearly independent tight constraints, and makes the counting argument work. Effect: Only violates the degree bound by an additive constant. B v = Lemma [LNSS]: If every vertex is of degree 5 when its degree constraint is present, then there is a half edge in a basic solution.

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Counting By linear independence +2 Theorem [LNSS]. Every basic solution has an edge with value at least 1/3 if every degree constraint has at least 5 edges. Induction Hypothesis: The root has 2 extra tokens.

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Iterative Relaxation Initialization: H =, f = f. While f 0 do: oFind a basic optimal solution, x, of the LP with function f. o (Rounding) Add an edge with x(e) 1/2 into H. o (Relaxing) Remove the degree constraint of v if v has degree 4 oUpdate the connectivity requirement f o Update degree bounds: set Bv:=Bv-1/2 if e is incident at v. Output H. An additive constant +3 A multiplicative factor 2 Theorem : There is a (2,2B+3)-approximation algorithm for the minimum bounded degree Steiner network problem.

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Additive Approximation Theorem [Lau,Singh,STOC 08]: There is a (2,B+O(r max ))-approximation algorithm for the minimum bounded degree Steiner network problem. Theorem [Lau,Singh,STOC 08]: There is a (2,B+3)-approximation algorithm for the minimum bounded degree Steiner forest problem.

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Minimum Bounded Degree Spanning Trees Theorem [Singh,Lau,STOC 07] There is an (1,B+1)-approximation algorithm for the minimum bounded degree spanning tree problem. Improves on the (1,B+2)-approximation of Goemans 2006

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Directed Connectivity (const, const)-approximation for certain directed Connectivity problems [LNSS STOC 07] [Bansal,Khandekar,Nagarajan,STOC 08]

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Exact Formulations Spanning Tree Arborescence Matroid intersection Perfect matching in general graphs Rooted k-out-connected subgraphs Submodular flows This method can be applied to prove LP formulations are exact. Limitation: not simple to prove the dual is integral.

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Approximation Algorithms Some NP-hard problems are variants of basic problems. General assignment (bipartite matching) Multicriteria spanning trees Partial vertex cover Prize collecting Steiner trees Degree bounded matroids [Király,L,Singh,08] Degree bounded submodular flows [Király,L,Singh,08]

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Open Problems TSP, ATSP? Other applications? Combinatorial algorithms? Connection to existing approaches?

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